Abstract
Modeling the energetic behavior of martensitic (phase transforming) materials usually leads to non quasiconvex energy formulations. For this reason, researchers often employ quasiconvex relaxation methods to improve the character of the formulation. Unfortunately, explicit expressions for the relaxed free energy density for multi-variant martensitic materials are typically not available. Thus, some researchers have employed a Reuβ-like convex lower bound, which neglects compatibility constraints, as an estimate on the free energy of mixing. To be confident with such a technique, one needs a measure of the quality of the lower bound. In this paper, we seek such a measure by comparing the Reuβ-like lower bound to an upper bound. The upper bound is constructed upon assumptions on the type of microstructures that form in such alloys. In particular, we consider lamination type microstructures which form by temperature- or stress-induced transformation in monoclinic and orthorhombic Copper-based alloys with cubic austenitic symmetry. Our results display a striking congruence of upper and lower bounds in the most relevant cases.
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References
Avellaneda, M., Milton, G.: Bounds on the effective elasticity tensor of composites based on two-point correlations. In: Hui, D., Koszic, T (eds) Proceedings of the ASME Energy-Technology Conference and Exposition ASME New York (1989)
Ball J.M. and James R.D. (1987). Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100: 13–52
Ball J.M. and James R.D. (1992). Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. London A 338: 389–450
Bhattacharya K. (2003). Microstructure of martensite: why it forms and how it gives rise to the shape-memory effect. Oxford University Press, New York
Govindjee, S., Mielke, A., Hall, G. J.: The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis. J. Mech. Phys. Solids 51, 763+ (2003)
Govindjee S. and Miehe C. (2001). A multi-variant martensitic phase transformation model: formulation and numerical implementation. Comput. Meths. Appl. Mech. Eng. 191: 215–238
Hall G.J. and Govindjee S. (2001). Application of a partially relaxed shape memory free energy function to estimate the phase diagram and predict global microstructure evolution. J. Mech. Phys. Solids 50: 501–530
Hall G.J. and Govindjee S. (2002). Application of the relaxed free energy of mixing to problems in shape memory alloy Simulation. J. Intelligent Mat. Sys. Struct. 13: 773–782
Hall G.J., Govindjee S., Šittner P. and Novák V. (2007). Simulation of cubic to monoclinic-II transformations in a single crystal Cu–Al–Ni tube. Int. J. Plasticity 23: 161–182
Kohn R. (1991). The relaxation of a double-well problem. Contin. Mech. Thermodyn. 3: 193–236
Otsuka K., Nakamura T. and Shimizu K. (1974). Electron microscopy study of stress induced acicular β′ − 1 Martensite in Cu–Al–Ni alloy. Trans. Natl Res. Inst. Met. 15: 200–210
Shield T. (1995). Orientation dependence of the pseudoelastic behavior of single crystals of Cu–Al–Ni in tension. J. Mech. Phys. Solids 43: 869–895
Sedlák P., Seiner H., Landa M., Novák V., Šittner P. and Mañosa L.l. (2005). Elastic constants of bcc austenite and 2H orthorhombic martensite in CuAlNi shape memory alloy. Acta Materialia 53: 3643–3661
Tartar L. (1990). H-measures, a new approach for studying homogenization, oscillation and concentration effects in partial differential equations. Proc. R. Soc. Edinb A 115: 193–230
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Govindjee, S., Hackl, K. & Heinen, R. An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations. Continuum Mech. Thermodyn. 18, 443–453 (2007). https://doi.org/10.1007/s00161-006-0038-1
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DOI: https://doi.org/10.1007/s00161-006-0038-1